Question

The total revenue function for a certain product is given by Requals=440440x ​dollars, and the total cost function for this product is Cequals=20 comma 00020,000plus+4040xplus+x squaredx2 ​dollars, where x is the number of units of the product that are produced and sold. a. Find the profit function. b. Find the number of units that gives maximum profit. c. Find the maximum possible profit.

Answer 1

(a)

Given:

Revenue Function is given by:

R440

Cost Function is given by:

C-20. 000 + 40x +

Profit Function is given by:

P = R-C = 440x _ (20.000 + 40x + x2) =ー2.2 + 400.-20. 000

So,

Profit Function is:

                       (1)

P =-x2 + 400x-20.000

(b)

Maximum Profit is got when

dP

So, differentiating (1) with respect to x and equating to 0, we get:

$-2x+400=0$

So,

x = 400/2

= 200

So,

The number of units that gives maximum profit is given by:

x = 200

(c)

Maximum Profit is given by:

P--2002 (400 × 200) _ 20. 000 = 40. 000+ 80. 000-20. 000 = 20. 000

So,

The maximum possible profit is given by:

20,000